I'm learning to play chess.
Twice we had a situation where the white player moved a piece that was blocking the black Queen from checking the white King. There was no move available to announce "Check".
What is the proper procedure here?
It's not technically a check since Black wasn't making a move to check White's king. White made an illegal move that placed his King under attack.
Black should mention that White made an illegal move, getting White to take the move back and play something else. If you're playing in a tournament, you could call the arbiter over and say White made an illegal move (often 3 illegal moves = forfeit).
The player of the white pieces has made an illegal move. Touch-move applies so he must take back his illegal move and make a legal move. If it is possible to make a legal move with the piece he touched then he must make a legal move with the touched piece. If there are no legal moves with the piece he touched then he may make any legal move. This is important because often making a legal move with the touched piece loses the game immediately.
Leaving the king exposed to check is an illegal move. Simply ask the player to make another legal move.